Blogging today from Last Drop Cafè in Claremont, CA. I think I found a new favorite drink: mint mocha made with soy milk.
Here are some basics of the conjugate and modulus of complex numbers.
Let z = x+i*y, with i=√1
Conjugate
Usually labeled "zbar" (z with a line over it), the conjugate is also labeled conj(z) and z*.
conj(z) = x  i*y
Modulus or Absolute Value
Not surprisingly, the modulus, also called the absolute value of the complex number z is defined as:
z = √(x^2 + y^2)
Properties
Let's explore some properties of the conjugate and modulus.
z + conj(z) = (x + i*y) + (x  i*y) = 2*x
z  conj(z) = (x + i*y)  (x  i*y) = 2*i*y
(conj(z))^2 = (x  i*y)^2 = x^2  2*i*x*y + (i*y)^2 = x^2  y^2  2*i*x*y
conj(z)^2 + z^2 = (x  i*y)^2 + (x + i*y)^2 = x^2  2*i*x*y  y^2 + x^2 + 2*i*x*y  y^2
= 2*(x^2  y^2)
conj(z) * z = (x  i*y) * (x + i*y) = x^2 + i*x*y  i*x*y  i^2*y^2 = x^2 + y^2 = z^2
which easily leads to: z = √(z * conj(z)) and
z1 * z2 = √(z1 * conj(z1) * z2 * conj(z2)) = √(z1 * conj(z1)) * √(z2 * conj(z2)) = z1 * z2
and:
z1/z2 = √((z1 * conj(z1))/(z2 * conj(z2))) = √((z1 * conj(z1))/√((z2 * conj(z2)) = z1/z2
Source: Wuncsh, David A. Complex Numbers with Applications. 2nd Edition. AddisonWesley Publishing Company. Reading, MA. 1994
A blog is that is all about mathematics and calculators, two of my passions in life.
Saturday, November 22, 2014
Complex Analysis: The Conjugate, the Modulus, and its Properties
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